OP, I really feel for you - you've taken the shitload of criticism you've received on the chin and pulled out all the positive advice to help you move forward.
Of course it's shit for the company. And of course it's not surprising that they want to address the situation. Totally understandable. No-one disputes that, including OP....
But let's just take a moment - this woman is looking after two very young children singlehandedly. Ex refuses to pay his share and has deliberately shirked his financial responsibilities as well as his practical ones. There have been multiple hospital admissions, not just sniffles and she hasn't been paid for the absence. She offered different working hours but this was declined (which is fair enough but she has tried to think of solutions...). She has just been through what sounds like a horrific set of personal circumstances and was a victim of DV. As a result of this DV and PND she had a year off work during which time Occ Health agreed she was unfit for work. After she split from her ex following the DV she lost her house, her car and her phone.
Fucking hell OP, you deserve a medal for still trying to keep going. No, it's not ideal for your employer and absolutely you need to try and explore some of the alternative suggestions. None of them will be an easy fix but with a bit of luck you might find something that works to keep you in employment and make you more reliable.
By the way, I didn't read your comment about making it in for half day as you being flippant - I thought you sounded like a woman who was desperately trying to fulfil her commitments and not knowing which way to turn.
I was a manager at a FTSE 100 company and also went through TUPE and I think there's reason for hope still. Make sure you have plans to cover any possible objections they might have, hopefully this is just the next stage of the process. It's true the TUPE process is ruthless and they'll try to get anyone out possible to reduce headcount - in my experience anyway.
Random thought - do you have any skills which would allow you to be self employed? If so, I wonder if you could start your own business? You could start working on it weekends and evenings to get it up and running and if it takes off, you could quit your day job. I have two young SEN children and I need flexibility. I couldn't commit to a FT day job any longer so I made the jump. Best thing I ever did.
Whatever you decide, I hope life starts throwing some better luck your way - no-one is blaming the employer at all but between the DV, losing your home, PND and very poorly children it really does sound like you've had a tough time.